Editing Low-dimensional topology (section)

==Three dimensions==
{{Main|3-manifold}}

A [[topological space]] ''X'' is a 3-manifold if every point in ''X'' has a [[neighbourhood (mathematics)|neighbourhood]] that is [[homeomorphic]] to [[Euclidean 3-space]].

The topological, [[Piecewise linear manifold|piecewise-linear]], and smooth categories are all equivalent in three dimensions, so little distinction is made in whether we are dealing with say, topological 3-manifolds, or smooth 3-manifolds.

Phenomena in three dimensions can be strikingly different from phenomena in other dimensions, and so there is a prevalence of very specialized techniques that do not generalize to dimensions greater than three.  This special role has led to the discovery of close connections to a diversity of other fields, such as [[knot theory]], [[geometric group theory]], [[hyperbolic geometry]],  [[number theory]], [[Teichmüller space|Teichmüller theory]], [[topological quantum field theory]], [[gauge theory]], [[Floer homology]], and [[partial differential equations]].  3-manifold theory is considered a part of low-dimensional topology or [[geometric topology]].

===Knot and braid theory===
{{Main|Knot theory|Braid theory}}

[[Knot theory]] is the study of [[knot (mathematics)|mathematical knot]]s. While inspired by knots that appear in daily life in shoelaces and rope, a mathematician's knot differs in that the ends are joined together so that it cannot be undone. In mathematical language, a knot is an [[embedding]] of a [[circle]] in 3-dimensional [[Euclidean space]], '''R'''<sup>3</sup> (since we're using topology, a circle isn't bound to the classical geometric concept, but to all of its [[homeomorphism]]s). Two mathematical knots are equivalent if one can be transformed into the other via a deformation of '''R'''<sup>3</sup> upon itself (known as an [[ambient isotopy]]); these transformations correspond to manipulations of a knotted string that do not involve cutting the string or passing the string through itself.

[[Knot complement]]s are frequently-studied 3-manifolds. The knot complement of a [[tame knot]] ''K'' is the three-dimensional space surrounding the knot. To make this precise, suppose that ''K'' is a knot in a three-manifold ''M'' (most often, ''M'' is the  [[3-sphere]]).  Let ''N'' be a [[tubular neighborhood]] of ''K''; so ''N'' is a [[solid torus]].  The knot complement is then the [[complement (set theory)|complement]] of ''N'',
:<math>X_K = M - \mbox{interior}(N).</math>

A related topic is [[braid theory]]. Braid theory is an abstract [[geometry|geometric]] [[theory]] studying the everyday [[braid]] concept, and some generalizations. The idea is that braids can be organized into [[group (mathematics)|group]]s, in which the group operation is 'do the first braid on a set of strings, and then follow it with a second on the twisted strings'. Such groups may be described by explicit [[presentation of a group|presentation]]s, as was shown by {{harvs|first=Emil|last=Artin|authorlink=Emil Artin|year=1947|txt}}.<ref>{{citation
 | last = Artin | first = E. | authorlink = Emil Artin
 | doi = 10.2307/1969218
 | journal = [[Annals of Mathematics]]
 | mr = 0019087
 | pages = 101–126
 | series = Second Series
 | title = Theory of braids
 | volume = 48
 | year = 1947}}.</ref> For an elementary treatment along these lines, see the article on [[braid group]]s. Braid groups may also be given a deeper mathematical interpretation: as the [[fundamental group]] of certain [[Configuration space (mathematics)|configuration space]]s.

===Hyperbolic 3-manifolds===
{{Main|Hyperbolic 3-manifold}}

A [[hyperbolic 3-manifold]] is a [[3-manifold]] equipped with a [[complete space|complete]] [[Riemannian metric]] of constant [[sectional curvature]] -1.  In other words, it is the quotient of three-dimensional [[hyperbolic space]] by a subgroup of hyperbolic isometries acting freely and [[Properly discontinuous action|properly discontinuously]].  See also [[Kleinian model]].

Its thick-thin decomposition has a thin part consisting of tubular neighborhoods of closed geodesics and/or ends that are the product of a Euclidean surface and the closed half-ray.  The manifold is of finite volume if and only if its thick part is compact.  In this case, the ends are of the form torus cross the closed half-ray and are called '''cusps'''. Knot complements are the most commonly studied cusped manifolds.

===Poincaré conjecture and geometrization===
{{Main|Geometrization conjecture}}

[[Thurston's geometrization conjecture]] states that certain three-dimensional [[topological space]]s each have a unique geometric structure that can be associated with them. It is an analogue of the [[uniformization theorem]] for two-dimensional [[surface (topology)|surface]]s, which states that every [[simply connected|simply-connected]] [[Riemann surface]] can be given one of three geometries ([[Euclidean geometry|Euclidean]], [[Spherical geometry|spherical]], or [[hyperbolic geometry|hyperbolic]]).
In three dimensions, it is not always possible to assign a single geometry to a whole topological space. Instead, the geometrization conjecture states that every closed [[3-manifold]] can be decomposed in a canonical way into pieces that each have one of eight types of geometric structure.  The conjecture was proposed by {{harvs|txt|authorlink=William Thurston|first=William|last= Thurston|year= 1982}}, and implies several other conjectures, such as the [[Poincaré conjecture]] and Thurston's [[elliptization conjecture]].<ref>{{citation
 | last = Thurston | first = William P. | authorlink = William Thurston
 | doi = 10.1090/S0273-0979-1982-15003-0
 | issue = 3
 | journal = [[Bulletin of the American Mathematical Society]]
 | mr = 648524
 | pages = 357–381
 | series = New Series
 | title = Three-dimensional manifolds, Kleinian groups and hyperbolic geometry
 | volume = 6
 | year = 1982| doi-access = free
 }}.</ref>